EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 251

6. Production

Definition 6.1. Production is the transformation of a vector of commodities into a different vector of commodities.

Production can be organized in either of two environments: • In hierarchical environments: within firms In market environments: between firms Important questions without clear answers: Why do firms exist? ◦ What determines their boundaries? ◦ Economists understand more about the market environment than • about the hierarchical environment.

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 252

We use a simple model of production within the firm in order to • generate supply curves.

Supply curves are analyzed within the market environment. • Definition 6.2. A production activity or y is a vector of inputs (negative) and outputs (positive) that describes a transformation of some goods into others.

Definition 6.3. A production set Y Rn is the set of all activities that are physically possible to⊂ implement. EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 253

Suppose the commodities are given by: • ice cream 2 cheese 1 ⎡ ⎤ ,andsupposey = ⎡ ⎤ Y. milk 6 ∈ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ − ⎥ ⎢ sugar ⎥ ⎢ 8 ⎥ ⎢ ⎥ ⎢ − ⎥ ⎣ ⎦ ⎣ ⎦ Then it is physically possible to convert 6 units of milk and 8 units • of sugar into 2 units of ice cream and 1 unit of cheese.

Real production in modern economies is extraordinarily • complicated.

Vectors of inputs and outputs would be very large. •

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 254

Think about the economics department at BU:

Inputs: • Professors, each one with different skills, training. Staff, again, each with different skills, training. Structure (this building) is specialized. Equipment, not too important.

Outputs: • Undergraduate Masters’ training Doctors’ Training Research of doctoral students Research of professors EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 255

How could you represent in a vector what we do in the economics • department?

How could you represent all the possible production activities in • the department, including those that have not been used here?

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 256

The economists’ model is extremely abstract! • The set Y , below, is the production set. • Y does not contain any points in Rn except 0. Why not? • + Think of the an activity vector y Y as an arrow from the origin • that indicates a decrease in the stocks∈ of inputs and an increase in the stocks of outputs.

n y . + Y 0

n . − EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 257

If a vector of initial resources, z0 , is added to each point in the • set Y ,aproduction possibility set (PPS) Y + z = y + z y Y is obtained. 0 { 0 | ∈ } In the diagram below, the activity y transforms the initial • resources z0 to the the vector of resources z1 .

z + Y 0 z1 y Production z0 Possibility Set 0

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 258

The PPS lies in Rn . • + Contains all the combinations of commodities that can be • obtained in the economy.

z corresponds to 0 in the original production set. • 0

z0 is the vector obtained if no production occurs. • EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 259

ThefollowingarecommonlyassumedtobepropertiesofY : Assumption 6.1. Continuity in production: Y is closed. Assumption 6.2. Inactivity is possible, but there are no free products: Y R+ = 0 . ∩ { } You can’t get outputs without using inputs. • But you can do nothing, so 0 Y • ∈

z n . + {y≤z} Y P(x) 0

x n . −

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 260

For any x Y , we define the higher-productivity set, P(x), by • P(x)= y∈ y Y , y x . { | ∈ ≥ } As compared with x,allthepointsinP(x) (except for x itself) either use smaller amounts of some inputs ◦ and/or yield greater amounts of some outputs ◦

z n . + {y≤z} Y P(x) 0

x n . − EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 261

Assumption 6.3. Limited productivity:

Given any x Y , the higher-productivity set P(x) is • bounded. [Missing∈ from M-C, but should be there.]

Assumption 6.4. Free disposability:

If z Y ,andy z ,theny Y . • ∈ ≤ ∈ It is always possible to waste inputs. • See graph. • Assumption 6.5. Irreversibility:

if y Y ,then y / Y . • ∈ − ∈ Can’t turn process around, produce inputs from outputs. Technical assumption: Labor, energy used in production, cannot be fully recovered.

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 262

Assumption 6.6. Convexity: Y is a convex set.

Strong assumption. • Rules out economies of scale. • Graph below violates convexity, shows increasing returns. •

n . + Y 0

n . − EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 263

Assumption 6.7. Additivity:

If x, y Y ,thenx + y Y . • ∈ ∈ Strong assumption. Rules out diseconomies of scale.

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 264

Assumption 6.8. Y is a convex cone: If x, y Y ,thenfor α, β > 0, αx + βy Y . ∈ ∈ Very strong assumption. • Implies additivity. • Mathematically, Y is closed • under linear combinations.

Implies constant returns to scale. • Y 0 EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 265

Replication argument: • Production sets have additivity property between two different production activities if they don’t interfere with one another. Production activities describe possible transformations, not resource constraints. If a production activity exits, then it ought to be possible to replicate (duplicate) it exactly, as many time as desired. = If y Y , then for any integer n, ny⇒ Y .∈ ∈ = Y has additivity. ⇒

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 266

Proposition 6.1. If 0 Y ,andY is convex and has additivity, then Y is a convex∈ cone. Proof. We have:

By convexity, if y Y , then for α [0, 1], • αy = αy +(1 α∈)0 Y . ∈ − ∈ For α > 1, find integer n α. • ≥ By additivity, ny Y ∈ By convexity αny Y . n ∈ = αy Y . ⇒ ∈ If x, y Y , then for α, β > 0, αx, βy Y . • ∈ ∈ By additivity: αx + βy Y . • ∈ EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 267

Suppose Y Rn is convex but not a convex cone. • ⊂

By assuming that there is a “hidden input,” z , we can create Y 0 • that is a convex cone.

Assume that each y Y , uses also one unit of z . • ∈ y For each y Y , set y 0 = . • ∈ 1 " − #

y n+1 Define Y 0 = α α 0, y Y R . • ( " 1 # ¯ ≥ ∈ ) ⊂ − ¯ ¯ Y is a convex cone. ¯ • 0 ¯

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 268

Definition 6.4. If there is only one output, q, and a number of inputs, denoted by the vector z ,thenifY is a closed set, has free disposability with 0 Y ,wedefine a production function f by ∈ q f(z)=max q Y ( ¯ " z # ∈ ) ¯ − ¯ ¯ ¯ EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 269

Definition 6.5. AproductionsetY exhibits increasing returns to scale if for any y Y and α > 1, there is a point y > ay with y Y . ∈ 0 0 ∈ If you increase the scale of production, then it is possible to • increase productivity.

Definition 6.6. AproductionsetY exhibits decreasing returns to scale if for any y Y and α (0, 1), there is a point y > ay with y Y . ∈ ∈ 0 0 ∈ If you decrease the scale of production, then it is possible to • increase productivity.

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 270

Definition 6.7. AproductionsetY exhibits nonincreasing returns to scale if for any y Y and α (0, 1), ay Y . ∈ ∈ ∈ It is always possible to decrease the scale of production. • Definition 6.8. AproductionsetY exhibits nondecreasing returns to scale if for any y Y and α > 1, ay Y . ∈ ∈ It is always possible to increase the scale of production. • Definition 6.9. AproductionsetY exhibits constant returns to scale if for any y Y and α > 0, ay Y . ∈ ∈ Nonincreasing and nondecreasing returns to scale. • EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 271

Proposition 6.2. Let f (x) with x Rn be a concave ∈ + production function. Define a hidden input z R+ and a production function ∈ 1 F (X, z) zf X . ≡ z µ ¶ Then F is concave, has constant returns to scale, and F (x, 1) f (x). ≡

The quantity • 1 x X ≡ z should be interpreted as the quantity of X per unit of the hidden input.

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 272

If you think of the hidden input z as management, this says that • each separate unit of managment can operate the production process described by f (x).

Therefore, a total of z units of management and X = zx units of • the original input gives us zf (x) units of output.

This suggests that if a production function has decreasing returns • to scale, it is because of a hidden input that is not increased as the scale of production increases. EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 273

Once the hidden input is introduced, the new production function • is concave and has constant returns to scale.

Equivalently, the set under the graph of the new production • function is a convex cone.

FXz(), fx() z

1 x, X

0

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 274

FXz(), fx() z

1 x, X

0 EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 275

Proof. We have 1 F (αX, αz)=αzf αX = αF (X, z) , αz µ ¶ which proves that F has constant returns to scale. In addition, letting X αX +(1 α) X and z αz +(1 α) z ,wehave 00 ≡ − 0 00 ≡ − 0 1 α 1 α F (X 00, z 00) z 00f X 00 z 00f X + − X 0 ≡ z ≡ z z µ 00 ¶ µ 00 00 ¶ αz 1 (1 α) z 0 1 = z 00f X + − X 0 . z z z z µ 00 00 0 ¶

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 276

And because αz (1 α) z0 + − =1, z00 z00 the concavity of f now implies:

αz 1 (1 α) z 0 1 F (X 00, z 00) z 00f X + − z 00f X 0 ≥ z z z z 00 µ ¶ 00 µ 0 ¶ 1 1 αzf X +(1 α) z 0f X 0 ≥ z − z µ ¶ µ 0 ¶ = αF (X , z )+(1 α) F (X 0, z 0) , − which proves that F is concave. EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 277

6.1 Profit Maximization and Cost Minimization

Definition 6.10. A firm is a price-taker if it assumes that all prices are fixed and that its own actions will have no effect on prices.

This makes sense when a firm is very small compared to the size • of the market.

For now, assume that firms are price-takers. • We also assume that there are n goods and that all production • sets Y are closed, contain 0,and have free disposability.

Let Rn p 0 . • ++ ≡ { À }

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 278

Definition 6.11. A firm’s profit-maximization problem (PMP) is: max py y Y y { | ∈ }

Definition 6.12. Suppose the PMP has a solution for every p 0. Then, the profit function π : Rn R is given by À ++ → π(p) max py y Y ≡ y { | ∈ } and the supply correspondence,by y(p) argmax py y Y . ≡ y { | ∈ }

Note: y is both the supply of outputs and the derived demand for • inputs. EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 279

Suppose there is one output q and a vector of inputs z ,with • scalar p > 0 the price of the output, and w 0 the vector of input prices. Then, the profit function is givenÀ by: π(p, w)=max pf(z) wz z 0 z { − | ≥ }

and z , the derived demand for inputs, by • z(p, w) = argmax pf(z) wz z 0 . z { − | ≥ } with the supply of output given by q(p, w)=f(z(p, w)).

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 280

From Kuhn-Tucker we know that • ∂f p = wj for zj(p, w) > 0 ∂zj and ∂f p wj for zj(p, w)=0 ∂zj ≤

This says that the value of the marginal product of an input must • equal its price if the input is used and must be less or equal to its price if the input is not used. EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 281

Proposition 6.3. If Y has nondecreasing returns to scale (which includes the case of constant returns to scale), then π(p) is not defined for some p 0. À Proof. Suppose y Y with y £ 0. ∈ Choose p 0 such that py > 0. • À

To do this choose pi small for yi < 0,largeforyi > 0.

For any α > 1, αy Y . • ∈ p(αy)=α(py) as α . • →∞ →∞ By definition • π(p) max py y Y , ≡ y { | ∈ } but py is unbounded for y Y , so the maximum doesn’t exist. ∈

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 282

Inwhatfollows,weassumethatπ(p) is defined for all p 0, • and that Y is closed, contains 0, and satisfies free disposal.À

Proposition 6.4. π is homogeneous of degree 1 and convex in prices. Proof. The proof of homogeneity is the same as for the expenditure function. For convexity:

Suppose p, p 0, α [0, 1] and p = αp +(1 α)p . • 0 À ∈ 00 − 0 For y y(p ) we have • 00 ∈ 00 π(p00)=p00y 00 = αpy 00 +(1 α)p0y 00 − απ(p)+(1 α)π(p0). [Why?] ≤ − EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 283

What is the economic interpretation of the convexity of the profit • function?

The average of profits at two different prices is at least as much as profits at the average prices. Profits are higher at extreme prices than at average prices.

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 284

6.2 The separating hyperplane theorem and duality in production.

Suppose you have a production set and a point outside the • production set.

How can you find a price vector that makes the outside point • more profitable than any point in the production set?

The separating hyperplane theorem answers that question. • EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 285

Definition 6.13. The set n Hp,x0 x x and x0 R ,p(x x0)=0 ≡ { | ∈ − } is a hyperplane of dimension n 1 orthogonal to p through the − point x0 .

A hyperplane is the union of all lines orthogonal to a fixed vector • andpassingthroughagivenpoint.

The budget frontier is a hyperplane orthogonal to the price vector • and passing through a any point x0 , such that px0 = w.

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 286

Proposition 6.5. (Separating Hyperplane Theorem) If B Rn is convex and closed, and if s / B, then ⊂ ∈ there is a hyperplane H of dimension n 1 such that B lies • on one side of H and s lies on the other;−

more precisely, there is some x0 and p = 0, such that for all • b B, pb px < ps. 6 ∈ ≤ 0 s p x0 B H px, 0 EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 287

Proof.

Let x0 be the closest point in B to s. • [How do we know that a “closest point” exists? See the problem set.]

set p s x0 • ≡ −

so that 0 < p p = p(s x0 ). • · − Therefore px < ps. • 0 Suppose for some y B, py > px . • ∈ 0 Then p(y x ) > 0. • − 0 Set x = αy +(1 α)x where α (0, 1). • 0 − 0 ∈ By convexity, x B. • 0 ∈

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 288 y s x ' p x B 0 H px, 0

We show that for small enough α, x 0 is closer to s than x0 is, a • contradiction:

s x 0 = s αy (1 α)x − − − − 0 =(s x ) α(y x ) − 0 − − 0 = p α(y x0 ) − − EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 289

so that for very small α we have, 2 s x 0 =(s x 0)(s x 0) k − k − − =(s x0 )(s x0 ) 2 − − +α (y x0 )(y x0 ) − − 2αp(y x0 ) − 2− < s x0 [Why?] k − k

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 290

Definition 6.14. The dual production set Y 0 of a production set Y is given by

Y 0 = y py π(p) for all p 0 , { | ≤ À } where π (p) is the profit function derived from Y . The dual production set is bounded by the lower envelope of the • hyperplanes defined by the various price vectors and points on the output axis as shown in the drawing below. y p4 p3 p2 π ()p4 p4y H 4 π ()p3 p3y

π ()p2 p p2y ′ 1 x Y H 3 H 2

π ()p 1 ≡ 0 H1 p1y EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 291

For many commonly used production functions, the profit function • is undefined at price vectors p that contain input prices equal to 0.

Example 6.1. Consider the production function q (`)=2√`, where ` is the input of labor.

Let the price of output be 1 and let w denote the wage rate, • so that π = 2√` w`. − The demand for labor ` (w) is unbounded at w = 0,and • π (0) is undefined.

But for all w > 0,wehave` (w)=1/w 2 ,and • π (w)=2/w 1/w = 1/w. −

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 292

Problem 6.1. Prove more generally that for well-behaved homogeneous production functions with decreasing returns to scale, π (p) is defined for all p 0 but is generally undefined at price vectors p that containÀ zero prices for one or more inputs.

For this reason, it makes sense to define the dual production set • in terms of p 0. À EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 293

Proposition 6.6. (Duality Theorem for Production Sets). Let Y 0 be the dual dual production set of Y .IfY closed, contains 0, has free disposability and is convex, then Y 0 = Y . The theorem says that just as you can derive the profit function • from the technology, so you can find the technology from knowledge of the profit function.

Proof.

First, suppose y Y . • ∈ Because π(p)=max py y Y ,wehave py π(p) for all p 0. { | ∈ } ≤ À Therefore y Y ,sothatY Y . ∈ 0 ⊂ 0

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 294

Now suppose y / Y . • ∈ δ ⎡ δ ⎤ Define vectors of the form δj ,whereδ > 0 and 1 is ≡ 1 − ⎢ ⎥ ⎢ − ⎥ ⎢ δ ⎥ ⎢ ⎥ the j th component. ⎣ ⎦ Set Y¯ = Y + α δj for all α 0. j j ≥ Note that as δ Pgoes to 0, Y¯ looks more and more like Y . Set δ small enough so that y / Y¯ . ∈ EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 295 p y

Y n .+ Y 0 Y

n .− Y

Apply separating-hyperplane theorem to Y¯ and y.

Find p and x0 such that px px0 < py for all x Y¯ . ≤ ∈ Note p 0. Why? À π(p) px0 < py ≤ y / Y . ∈ 0

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 296

Proposition 6.7. (From M-C 5.C.1. pp. 138.) Suppose Y is closed, contains 0, and has the free-disposal property, and suppose π is defined for all p 0 . À i). the supply (and derived demand) relation y( ) is homogeneous of degree 0. ·

ii).IfY is convex, then for each p,thesety(p) is convex. iii).IfY is strictly convex, then y( ) is a function. · iv). (Hotelling’s Lemma) If y(p) is a single point, then the profitfunctionπ is differentiable at p and y(p)= π(p). ∇ v).Ify( ) is a function and is differentiable at p, then ∂y(p·)/∂p is a symmetric positive-semidefinite matrix with ∂y(p) p =0. ∂p EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 297

The proofs are similar to the corresponding proofs for the • expenditure function.

There are no income or wealth effects with supply and derived • demand.

Inputs are purchased, converted to outputs and then sold, all • simultaneously.

No wealth is required in these models. •

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 298

Proposition 6.8. (Law of Supply) Suppose y( ) is a supply · (derived demand) function. Then for any two price vectors p0 and p 0,we have À (p0 p)(y(p0) y(p)) 0. − − ≥

Proof. Because y(p) maximizes profits at prices p,weknowthat

p0y(p0) p0y(p) ≥ and py(p) py(p0) ≥

If only one price changes, then the corresponding quantity • changes in the same direction

Because ∂y(p)/∂p is a symmetric positive-semidefinite matrix, • own-price elasticity of supply is positive, and that of derived demand is negative (why?). EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 299

SupposewearegivenaproductionsetY and a price vector p. • Sequential maximization is possible. • We can find a short-run profit function holding capital fixed, ... and from that we can find a long-run profit function allowing capital to vary.

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 300

Example 6.2. Suppose the technology is defined by q √ Y = ⎧⎡ k ⎤ ¯ k, ` 0 and q k`⎫ , − ¯ ≥ ≤ ⎨⎪ ` ¯ ⎬⎪ ⎢ − ⎥ ¯ ⎣ ⎦ ¯ and suppose p, r ⎩⎪and w, are¯ the prices of q, k and⎭⎪ `.Thenfind the short- and long-run pro¯ fit functions if k is fixed in the short run.

The short-run passive profit function is: • π (p, r, w k)=max p√k` rk w` | ` − − [Why don’t we have to maximizen over q?] o EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 301

The foc is • ∂π p k 0= = w ∂` 2r` − so that k w 2 ` = . 4 p ,µ ¶

Substituting into the passive profit function gives • π (p, r, w k)= p√k` rk w` k w 2 | − − `=4 ( p ) h p2 i . = r k. 4w − µ ¶

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 302

What happens if • p2

The long-run profit function is • p2 π (p, r, w)=max r k = ??? k 4w − µ ¶ EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 303

6.3 Cost Minimization

If some commodities are always inputs and others are always • outputs, then we can break the profit maximization problem into two parts:

The minimization of costs over input vectors z conditional on the output vector q. The maximization of profits over all output vectors q.

Similar to short-run and long-run profits, above. • For simplicity we assume that there is only one output q. • Let q = f (z ) be the production function, p, the price of the • output, and w, the vector of input prices.

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 304

Definition 6.15. The passive cost function is defined by CP(z w)=wz . |

The passive cost function adds up the cost of the shopping list. • It is of no use unless you know z . • We would like to know how much it cost to produce a given • quantity of output.

Definition 6.16. The cost minimization problem (CMP) is

min CP(z w) z | s.t. f (z ) q ≥ z 0 ≥ EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 305

Definition 6.17. The cost function c(w q) is the solution of CMP, that is | c(w q)=min wz f(z) q . | z 0 { | ≥ } ≥

Definition 6.18. The conditional derived-demand function z (w q) solves the CMP, that is | z(w q)=argmin wz f(z) q . | z 0 { | ≥ } ≥

Proposition 6.9. The profit function is given by π(p, w)=max pq wz(w q) q { − | }

Proof. Apply sequential maximization.

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 306

If w is assumed fixed and q is assumed variable, then we use: • Definition 6.19. The total cost function is defined by C (q)=c(w q). | Definition 6.20. The marginal cost function is defined by MC(q) C (q). ≡ 0 Definition 6.21. The average cost function is defined by AC(q) C (q)/q. ≡

Proposition 6.10. Suppose q ∗ > 0 minimizes AC(q). Then MC(q ∗)=AC(q ∗).

Why? • EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 307

Proposition 6.11. (M-C.5.C.2) Suppose that c(w, q) is the cost function of a single-output technology Y , closed with free disposal and with production function f (z ), and suppose that z (w q) is the associated derived-demand correspondence. Then|

i). c(w q) is homogeneous of degree 1 in w and nondecreasing in q.|

ii). c(w q) is a concave function of w. | iii). (Duality theorem for cost functions) If z z 0, f (z ) q is convex for every q,then { | ≥ ≥ } Y = ( z, q) wz c(w q) for all w 0 . { − | ≥ | À }

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 308 iv). z (w q) is homogeneous of degree 0 in w. | v).If z z 0, f (z ) q is convex, then z (w q) is convex. If z{ z| ≥0, f (z ) ≥q is} strictly convex then|z (w q) is {single| ≥ valued. If≥ z }z 0, f (z ) q is strictly convex| for all q,thenz is a{ function.| ≥ ≥ } vi). (Shepard’s Lemma) If z (w q) is a function, then c(w q) is differentiable, and | | c(w q) z(w q). ∇w | ≡ | EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 309

vii).Ifz (w q) is differentiable, then | ∂z(w q) ∂2c(w q) | | ∂w ≡ ∂w2 so that ∂z (w q)/∂w is a symmetric negative-semidefinite matrix with | ∂z(w q) | w =0. ∂w viii).Iff (z ) is homogeneous of degree 1 (has constant returns to scale), then so are c(w q) and z (w q) in q. | | ix).Iff (z ) is concave, then C (q) is a convex function of q (MC is increasing).

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 310

Example 6.3. Find conditional derived demand, the cost function, supply and the profit function when α 1 α Q F(K , L) AK L − ,where0 < α < 1,withprices p, r≡, w. ≡

Both capital and labor must be used to obtain positive output. • Therefore • ∂F ∂F ∂K = ∂L r w AαL1 α A (1 α) K α − = − 1 α α rK − wL wL rK = 1 α α − α w K = L. 1 α r − This shouldn’t be surprising, because with a Cobb-Douglas • production function, the share of expenses for labor and capital must be proportional to the exponents. EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 311

Substituting into the production function yields the conditional • derived demands: α w α Q = A L 1 α r µ − ¶ 1 1 α α r α L∗ = − Q A α w µ ¶ µ ¶ 1 α 1 α 1 α − w − K ∗ = Q. A 1 α r µ − ¶ µ ¶ Therefore the cost function is • C (r, w, Q)=rK ∗ + wL∗

α 1 α C (r, w, Q)=Ar¯ w − Q where α 1 α 1 1 α 1 α − A¯ = − + . A α A 1 α µ ¶ µ − ¶

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 312

We have • α 1 α AC = MC = Ar¯ w −

If p < AC , nothing will be produced and profits are zero. • If p > AC , production will be unbounded and so will profits. • EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 313

6.4 Do firms behave according to the neoclassical model?

Profit maximization may share some of the same problems as • utility maximization.

The maximization problem is very difficult Firms don’t know their costs. ◦ Firms don’t know which production activities are possible. ◦ Questions of psychology Entrepreneurs are biased by optimism and hubris. ◦ Time inconsistencies ◦

EC 701, Fall 2005, Microeconomic Theory October 31, 2005 page 314

Firms have problems that individuals do not. • Principal/agent questions. How to get workers to work hard? ◦ How to get managers to pursue the interests of the firm’s ◦ owners? Problems of managerial control of firms. · Private benefits. · Owners of firms may maximize their utility rather than profits. Risk aversion. ◦